The functional constraints method: Application to non-linear delay reaction-diffusion equations with varying transfer coefficients

We present a number of new simple separable, generalized separable, and functional separable solutions to one-dimensional non-linear delay reaction-diffusion equations with varying transfer coefficients of the formut=[G(u)ux]x+F(u,w),where u=u(x,t) and w=u(x,t−τ), with τ denoting the delay time. All of the equations considered contain one, two, or three arbitrary functions of a single argument. The generalized separable solutions are sought in the form u=∑n=1Nφn(x)ψn(t), with φn(x) and ψn(t) to be determined in the analysis using a new modification of the functional constraints method. Some of the results are extended to non-linear delay reaction-diffusion equations with time-varying delay τ=τ(t). We also present exact solutions to more complex, three-dimensional delay reaction-diffusion equations of the formut=div[G(u)∇u]+F(u,w).Most of the solutions obtained involve free parameters and so may be suitable for solving certain problems as well as testing approximate analytical and numerical methods for non-linear delay PDEs.